Universal results for the outlier eigenpair of oriented random graphs

Random matrices are formidable tools to build simple models of networked complex systems composed of large assemblies of interacting entities, being them proteins, neurons or individuals. The central idea consists in modeling the adjacency matrix of a complex network by a random matrix drawn from a certain ensemble. With our increasing power to collect, manipulate, and visualize large amounts of data, it has become evident that classic random matrix ensembles completely fail in describing the pattern of interconnections in real-world networked structures, since one element of a network usually influences a finite number others. As a consequence, the associated random matrix is sparse, namely it contains a large number of zero entries. This poses many difficulties, as most of the techniques and ideas of random matrix theory only applies to invariant or Gaussian random matrices, where all entries of the matrix are typically nonzero. What makes the theory of sparse random matrices so challenging is that there is no general recipe, and one has to often tailor a specific approach to deal with a specific statistical observable. In this talk I will introduce a novel analytical approach to study the statistics of the outlier eigenpair of sparse random matrices. In the case of the adjacency matrices of oriented random graphs, I will show that our theory yields exact analytical results for the outlier eigenvalue, the first two moments of the corresponding eigenvectors, the support of the spectral density, and the spectral gap. I will show that these spectral observables have an universal behavior, since their analytical expressions hold for a broad class of oriented random graphs, independently of their structural details.